Barnes
G
-Function (Double Gamma Function)
G
(
z
+
1
)
=
Γ
(
z
)
G
(
z
)
,
G
(
1
)
=
1
,
G
(
n
)
=
(
n
-
2
)
!
(
n
-
3
)
!
⋯
1
!
n
=
2
,
3
,
…
G
(
z
+
1
)
=
(
2
π
)
z
2
exp
(
-
1
2
z
(
z
+
1
)
-
1
2
γ
z
2
)
×
∏
k
=
1
∞
(
(
1
+
z
k
)
k
exp
(
-
z
+
z
2
2
k
)
)
Ln
G
(
z
+
1
)
=
1
2
z
ln
(
2
π
)
-
1
2
z
(
z
+
1
)
+
z
Ln
Γ
(
z
+
1
)
-
∫
0
z
Ln
Γ
(
t
+
1
)
ⅆ
t
The
Ln
's have their principal values on the positive real axis and are
continued via continuity.
When
z
→
∞
in
|
ph
z
|
≤
π
-
δ
(
<
π
)
Ln
G
(
z
+
1
)
∼
1
4
z
2
+
z
Γ
(
z
+
1
)
-
(
1
2
z
(
z
+
1
)
+
1
12
)
Ln
z
-
ln
A
+
∑
k
=
1
∞
B
2
k
+
2
2
k
(
2
k
+
1
)
(
2
k
+
2
)
z
2
k
see
Ferreira and López(2001)
. This reference also provides bounds for the error term. Here
B
2
k
+
2
is the Bernoulli number, and
A
is Glaisher's constant, given by
A
=
ⅇ
C
=
1.28242 71291 00622 63687
…
where
C
=
lim
n
→
∞
(
∑
k
=
1
n
k
ln
k
-
(
1
2
n
2
+
1
2
n
+
1
12
)
ln
n
+
1
4
n
2
)
=
γ
+
ln
(
2
π
)
12
-
ζ
′
(
2
)
2
π
2
=
1
12
-
ζ
′
(
-
1
)
and
ζ
′
is the derivative of the zeta function
For Glaisher's constant see also
Greene and Knuth(1982)
(p. 100).
